Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $k = \dfrac{6r - 42}{9} \div \dfrac{9(r - 7)}{-6} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{6r - 42}{9} \times \dfrac{-6}{9(r - 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ (6r - 42) \times -6 } { 9 \times 9(r - 7) } $ $ k = \dfrac {-6 \times 6(r - 7)} {9 \times 9(r - 7)} $ $ k = \dfrac{-36(r - 7)}{81(r - 7)} $ We can cancel the $r - 7$ so long as $r - 7 \neq 0$ Therefore $r \neq 7$ $k = \dfrac{-36 \cancel{(r - 7})}{81 \cancel{(r - 7)}} = -\dfrac{36}{81} = -\dfrac{4}{9} $